Normal forms and internal regularization of nonlinear differential‐algebraic control systems

Abstract

In this article, we propose two normal forms for nonlinear differential‐algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, that is, when there exists a state feedback transforming the DACS into a differential‐algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs.

Modeling, analysis and control of robot–object nonsmooth underactuated Lagrangian systems: A tutorial overview and perspectives

“…The switching-DAE approach has been widely used in biped locomotion control [30]. In a broader context, each equality-constrained subsystem can be controlled using techniques derived for DAE [127,368]. However, one should not forget the fact that the switching rule is given by the CP (which is a controlled LCP), secondly switchings which involve an increase in the system's dimensions (in other words, the number of active constraints is augmented) are necessarily accompanied, in the case of (1), i.e., of relative degree 2 systems, by impacts (this is sometimes neglected [369]), thirdly it is not always easy to get an independent set of generalized coordinates that is valid globally for each DAE (this is easy for biped robots).…”

Impulse-free jump solutions of nonlinear differential–algebraic equations

“…to the book [25] for the formal definitions of such notions. We now recall some basic notions and results from the geometric analysis of the existence and uniqueness of C 1 -solutions for nonlinear DAEs (see e.g., [1,17,[26][27][28][29]).…”

Stability analysis of switched nonlinear differential-algebraic equations via nonlinear Weierstrass form

“…where T x M k ⊆ R n denotes the tangent space of the submanifold M k at x ∈ M k . The last approach is called the geometric reduction method [16], [14], [6], [3], [7] and it is proved in [6], [3] that the limit M * := M n of the sequence M k coincides with the consistency space C of Ξ.…”

“…Nevertheless, obtaining a fully-decoupled normal form for nonlinear DAEs has been an open problem for decades, some efforts have been made for studying normal forms of nonlinear DAEs, see e.g. [2], [7], [3].…”

“…Note that the assumption is simplified to adjust the purpose of the present paper, in general, M k is a subset of M k−1 and we may need to take a neighborhood U k ⊆ X such that M k ∩U k is a smooth connected embedded submanifold, see[3],[7] 2 A set S ⊆ R n is called a star domain if for each x ∈ S, it holds that λx ∈ S for all λ ∈ [0,1] …”